A short proof of the martingale convergence theorem

Abstract

The martingale convergence theorem is first proved for uniformly integrable martingales by a standard application of Doob’s maximal inequality. A simple truncation argument is then given which reduces the proof of the L 1 {L^1} -bounded martingale theorem to the uniformly integrable case. A similar method is used to prove Burkholder’s martingale transform convergence theorem.